You're lying about me again:

TonesinDeepFreeze has been asserting that "2+1" denotes the same object as "3" does, in a similar way. They very clearly each signify something different. The only attempt by Tones, to support this conclusion with a premise, was a vague reference to extensionality. — Metaphysician Undercover

(1) I didn't make "vague references". Indeed, I posted an explanation of the notion of exentionsality vs. intensionality. And I gave references in the literature for you to read about it. Moreover, even if I had not done that, it is still the case that the notion of extensionality vs. intensionality is a well known basic notion in the philosophy of mathematics and philosophy of language. The fact that you're ignorant of such basics of the subject is not my fault and doesn't make my reference to them "vague", and especially not when I gave explanation and additional references in the literature anyway.

(2) I posted multiple times that proving that '2+1' and '3' denote the same object is the basis on which we justify claiming that they do. Or, for a better example (since the equation '3 = 2+1' has such a trivial proof), we say '6-3' and '2+1' denote the same object because we prove that they do.

Stop lying about me.

Again on this point:

we do not have the premises required to conclude that they denote the same object. Therefore your conclusion that they denote the same object is fallacious. — Metaphysician Undercover

Again, this is conflating denotation is evidence, proof, demonstration and knowledge.

We conclude that two different names denote the same object by proving they do.

'2+1' and '6-3' have the same denotation. Of course, to legitimately assert that '2+1' and '6-3' requires first having grounds for the assertion, such as mathematical proof of the equation:

2+1 = 6-3.

That's what mathematicians do; they prove formulas, including equations. When the equation is proven, then we are justified in claiming that '2+1' and '6-3' denote the same object.

In any and every proposition about "Henry Fonda," we could substitute "the father of Peter Fonda" without changing the truth value. — aletheist

That's correct in an extensional context, but not in an intensional context:

Suppose Alice doesn't know that Henry Fonda is the father of Peter Fonda. Then consider these two sentences:

(1) Alice knows that Henry Fonda is Henry Fonda.

(2) Alice knows that Henry Fonda is the father of Peter Fonda.

But (1) is true (since Alice knows that identity is reflexive), while (2) is false even though (2) just substitutes 'the father of Peter Fonda' for 'Henry Fonda'.

'knows that' creates an intensional context in which substitution may not preserve truth values.

I have given you copious explanation. There's no point in me composing more explanation when it is better said anyway at the sources I offered you.

You are not recognizing the distinctions between terms and statements, terms and arguments, reference and inference, extension and intension, denotation and connotation.

Rather than composing lessons for you on the subject, I recommend:

https://plato.stanford.edu/entries/frege/ [Section 3]

https://plato.stanford.edu/entries/logic-intensional/

'Introduction To Mathematical Logic' pgs 1-68 - Alonzo Church

what in the expression "2+1" denotes that there is a result — Metaphysician Undercover

I addressed that already. The term '2+'1' denotes the value of the function + applied to the argument pair 2 and 1. It denotes the result of any computation of the function applied to those arguments. But it does not say within itself "there is a result". A term denotes an object (in this case, the object is the value of the function for the arguments, or the result of a computation of the function); a term is not itself a statement that there is a value or result.

'The father of Peter Fonda' denotes the value of the function (call it 'the father of function') applied to the argument Peter Fonda. That value is Henry Fonda.

'The father of Peter Fonda' does not itself denote the claim that there is value for the function.

A function is a process. — Metaphysician Undercover

As I mentioned, I'm not going to go along with your undefined terminology 'process'. Instead I'll use 'operation' (meaning a function) and 'procedure' (meaning an algorithm).

Grand Minnow kept insisting that "2+1" does not signify a process. That's why I say there is inconsistency. — Metaphysician Undercover

An inconsistency would be:

"'2+1' does not denote a process" and '''2+1' does denote a process".

But I never claimed that '2+1' denotes a process or operation or function or procedure.

So there is no inconsistency.

For about the seventh time now: '2+1' denotes the value of the function + applied to the argument pair 2 and 1. '2+1' does not denote a procedure nor a process (whatever vague notion of 'process' you probably have in mind. 2+1 denotes the RESULT of the procedure, not the procedure itself.

But clearly an "operation" or "function" is a process, and that's what is signified with "+". — Metaphysician Undercover

Yes, '+' denotes a function. But '2+1' does not denote the function. For about the eighth time now: '2+1' denotes the VALUE of the function for the arguments 2 and 1.

And rigorously a function is not a procedure. A function is a relation such that no member of the domain is related to more than one member of the range.

The "value" of the function is not signified, because it must be figured out by carrying out the operation — Metaphysician Undercover

I addressed that already. You skipped what I wrote about that and instead adduced an analogy that doesn't apply:

If I say add some sugar to water, and bring it to a boil — Metaphysician Undercover

Yes, that is a description of a process in the sense of a procedure (though, of course, only by analogy and not a mathematical procedure), and the result is syrup. But '2+1' is not a description of a procedure. A description of a procedure would be a statement of the recursive instructions for addition (and specifically for the inputs 2 and 1). '2+1' is not the name of a sequence of instructions. (Granted, in constructive mathematics, roughly put, there are notions of mathematical objects, such as numbers, being a construction. But we're not in that context, or to get to that context, you would need to understand a lot more about it.)

The value is not signified, the "procedure for determining the value" is what is signified. Do you agree? — Metaphysician Undercover

No, and see above and my previous posts.

Do you even know what the use-mention distinction is?
— TonesInDeepFreeze

Of course I do. In philosophy we use a different convention. I use " " to signify a concept rather than a physical thing. — Metaphysician Undercover

Use-mention is a convention in philosophy. It's not a different convention from that used in mathematics. And "concept vs. physical thing" is not it at all.

I'm trying to conform to your convention but I'm a bit sloppy and missed one. Call it a typo. — Metaphysician Undercover

You've been doing it over and over again. Not just typos.

do you understand the difference between a procedure (function, or operation), and an object? — Metaphysician Undercover

I am the one who has been harping on that difference.

Have you ever seen a ledger? Every account must be stated and balanced. Call it redundancy if you want, but there must be no room for error. — Metaphysician Undercover

You're babbling and again skipped my point. As I said, the accountant doesn't have to write '=?' or 'x' to add the numbers.

You just told me there is an infinite number of ways to say "2+1" — Metaphysician Undercover

Wrong. Use-mention again. I said there are infinitely many ways to denote 2+1, and '2+1' is one of those ways.

Why is "3+1" not just another one of the infinite ways of saying "2+1"? — Metaphysician Undercover

Another use-mention error by you.

you said things like "500+ 894+202" denote "a procedure for determining the value of a function applied to an argument" — Metaphysician Undercover

You did it again. You fabricated what I said.

You have it completely backwards what I said.

Your claim that "500+ 894+202" represents the value — Metaphysician Undercover

Yes, THAT is what I said. I said '500+ 894+202' denotes the value, not the procedure for determining the value.

This is probably around twenty times I've said it.

Now you got it right. So stop also fabricating that I said the opposite.

Two points. (1) You stated falsely about my motivation. (2) You outright fabricated a quote to make me look like I said the exact opposite of what I have been saying.

(1)

Now you're pretending to be someone else, so that your inconsistency is not so glaring. — Metaphysician Undercover

* I said at the very top of my last post that I am Grandminnow. I'm not pretending anything. And you lied by claiming that my motivation is to make anything less glaring.

* You have not shown any inconsistency.

(2)

You distinctly said "2+1" refers to "a procedure for determining the value of a function applied to an argument." — Metaphysician Undercover

I distinctly did NOT say that. And you put that misrepresentation in quotes to fabricate something I did not say.

I said that one notion of a process is that of a procedure for determining the value of a function applied to an argument.

And I say '2+1' does NOT denote a function nor an operation nor a process nor a procedure for determining the value of a function applied to an argument.

So stop right here. Go back to what I actually posted and see that I did not say, as you pretended to quote me, "'2+1' [refers to] a procedure for determining the value of a function applied to an argument."

If you don't then recognize that you fabricated a quote, then I won't know that I not talking with someone plainly dishonest and/or with real cognitive problems.

[This is GrandMinnow with a different name.]

all you've offered is inconsistency — Metaphysician Undercover

Again, you argue by mere assertion while evading the replies given to you. This is a paraphrase:

You: You are inconsistent.

Me: To be inconsistent is to claim or imply a contradiction, which is a statement and its negation. You have not shown that I've implied both a statement and its negation.

You: You are inconsistent.

What denotes that the operation has a result? — Metaphysician Undercover

That is just so daft!

The term itself doesn't denote that it has a result.

'+' is an operation symbol. An operation is a function. The usage "result of an operation" is an informal way of referring to the value of the function for the arguments. Every function has a value for arguments in its domain. That is, every function has a result when applied to arguments in its domain.

Would you agree that a finite operation is distinct from an infinite operation — Metaphysician Undercover

For a while, in order not to split hairs, I went along with your term 'process', even though you have not defined it. That was okay for a while, but I was concerned that it would cause confusion, since there are actually two different notions: (1) a function. (2) a procedure for determining the value of a function applied to an argument. (I did touch on this earlier.)

So I'm not going to go along with your undefined terminology 'process'. Instead I'll use 'operation' (meaning a function) and 'procedure' (meaning an algorithm).

As I touched on before, there are:

(1) procedures that terminate, (2) procedures that do not terminate, and (3) supertasks that are not finite but terminate. (1) and (2) are mathematical, and (3) is philosophical (or I am not versed in whatever mathematics there might be about it).

2+1 demotes exactly nothing then — Metaphysician Undercover

Use-mention! Please, I've pointed out a dozen times: use-mention!

2+1 is a number. '2+1' is a term that denotes.

Do you even know what the use-mention distinction is? I doubt you care to know even the most basic considerations in the subject of mathematical language.

There are an infinite number of ways to refer to the number 3. That doesn't mean they don't refer! Your argument is so daft!

infinity of different things referred to by on signification — Metaphysician Undercover

I pointed out in my last post that you have this exactly backwards. And you just repeat yourself again.

what makes 2+1 different from 3+1 then? — Metaphysician Undercover

Are you serious? Are you trolling?

The accountant writes out 500+ 894+202=?, or x — Metaphysician Undercover

No he doesn't. If he does, he's wasting precious billable seconds. Instead, he just goes ahead to add the numbers.

it were like you say, that "500+ 894+202" already says 1596, then the accountant would not have to sum up the numbers — Metaphysician Undercover

No, the term '500+ 894+202' already denotes 1596. It's just that the accountant doesn't know that until he performs the addition. The term doesn't start denoting only upon the knowledge of the account. The term doesn't spring into denotation every time some human being or computer somewhere in the world does a calculation.

equations contain unknowns — Metaphysician Undercover

You have no idea how daft that is.

An equation might contain only constants such as:

2+1 = 3

or it might contain a combination of constants and variables such as:

2+1 = x

or it might contain only variables such as:

x+ y = z

If "2+1" says sqrt(9), how is there any need to work out any equivalencies? — Metaphysician Undercover

Because we might not know that 2+1 = sqrt(9). The fact that at some point we don't yet know that 2+1 = sqrt(9) doesn't mean that it wasn't true all along. Obviously!